Details

Here are the steps for Chebyshev differentiation via the fast Fourier transform (FFT) [1] algorithm:

1. Take the Chebyshev–Gauss–Lobatto points given by with . These points are the extrema of the Chebyshev polynomial of the first kind, .

2. Calculate and form the vector .

3. Calculate , the real part of the and of .

4. Compute , where ; then calculate where is the inverse fast Fourier transform.

5. Use the vector to evaluate (i.e., the approximate derivative calculated at the Chebyshev–Gauss–Lobatto points) for :

for ,

,

and .

Reference

[1] L. N. Trefethen, Spectral Methods in MATLAB, Philadelphia: SIAM, 2000.